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Optimization of Linear Placements for Wirelength Minimization with Free Sites A. B. Kahng, P. Tucker, A. Zelikovsky (UCLA & UCSD) Supported by grants from Cadence Design Systems, Inc. http://vlsicad.cs.ucla.edu

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Outline Single-Row Problem Cell Cost Function Exact Algorithms for Single-Row Problem –Dynamic Programming Algorithm –Prefix Algorithm –Clumping Algorithm Swapping Heuristic for Cell Ordering Experimental Results Conclusions and Future Directions

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Single-Row Problem fixed cells movable cells C1C2C3C4C5C6C7

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Single-Row Problem Given –single cell row with n movable cells C[i] with fixed left-to-right order (but variable positions) and integer lattice of k sites (k > n) –m signal nets N [j] containing fixed cells from other rows Find –non-overlapping placement of n movable cells at k sites minimizing the total bounding-box half- perimeter of all m nets.

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Net with Movable and Fixed Cells single row with movable cells fixed cells net N span (N) fl(N) fr(N) ml(N) mr(N) fixed_span (N) minimize

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Cell Cost Function Cell cost function of C[i] = sum over all nets N of contributions of C[i] to span(N) - fixed_span(N) Given position x of cell C[i], cell cost function = cost[i](x) = max{mr(N) - fr(N),0} C[i] = rightmost movable on net N + max{fl(N) - ml(N),0} C[i] = leftmost movable on net N Total # linear pieces 2 #pins = 2 #nets = 2m

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Properties of Cell Cost Function Cost function of multi-pin cell is piecewise-linear and convex fr(1)fl(2)fl(3)fr(3)fr(2)fl(4)fr(4) minimum segment (point) If each cell is placed in its minimum segment, total bounding box half-perimeter is minimized

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Exact Algorithms for Single-Row Problem Dynamic Programming Algorithm –based on pre-computed cell cost functions Prefix Algorithm –based on piecewise-linearity of cell cost function Clumping Algorithm –based on convexity of cell cost function

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Dynamic Programming Algorithm Optimum constrained prefix placement P[i,j] of C[1],..., C[i] subject to C[i] being left of site s[j] P[i,j] is selected from P[i,j-1] and P[i-1,j-w [i-1] ] extended by C[i] at s[j] w [i-1] = width of C[i-1] Cost of prefix placement increased by cost[i](s[j]) Runtime = (i-range) (j-range) = n (k - w[i]) O(n 2 )

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Dynamic Programming Algorithm C[i-1] C[i] s[j] s[j-w[i-1]] C[i] s[j-1] P[i,j] has either: C[i] exactly at s[j] (extend P[i-1,j-w[i-1]]) or C[i] to left of s[j] (use already-computed P[i,j-1])

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Prefix Algorithm Prefix cost function pcost[i](x) = optimal placement cost of first i cells subject to C[i] being left of x pcost[i](x) is piecewise-linear decreasing Each linear segment is tuple = [a,b, min,max] Computing pcost[i] from pcost[i-1] and cost[i] merging sorted tuple sequences of sizes j*
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"name": "Prefix Algorithm Prefix cost function pcost[i](x) = optimal placement cost of first i cells subject to C[i] being left of x pcost[i](x) is piecewise-linear decreasing Each linear segment is tuple = [a,b, min,max] Computing pcost[i] from pcost[i-1] and cost[i] merging sorted tuple sequences of sizes j
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Prefix Algorithm pcost[i-1] x cost cost[i] pcost[i]

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Clumping Algorithm For each cell C[i], find –list of coordinates where cost[i] changes slope –C[i]’s minimum segment To each cell in order, apply PLACE(C[i]) Output positions of cells Procedure PLACE(C[i]) if C[i-1] and C[i] cannot be both in their minimum segments then COLLAPSE(C[i-1],C[i]) and PLACE(C[i-1]) else place C[i] at leftmost optimal available position

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Clumping Algorithm Procedure COLLAPSE(C[i-1],C[i]) – shift positions from the list of C[i] by width(C[i-1]) – merge the list for C[i] with the list for C[i-1] – find minimum segment for merged list – width(C[i-1]) = width(C[i-1]) + width(C[i]) – delete cell C[i] Using red-black trees for representation of cell lists, achieve runtime = O(m log m), m = # nets

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Clumping Algorithm clumped cell clumped cell optimal positions for cells directions to minimum segments of individual cells

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Swapping Heuristic for Cell Ordering Cell-Ordering Problem = the Single-Row Problem where the left-to-right order of cells is not fixed Swapping Heuristic Repeatedly iterate down the row until no pairs swap: –for every adjacent pair of cells that overlap or change order when placed at respective min points, swap their order if placement cost improves

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Experimental Results

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Conclusions First optimal algorithms for single-row cell placement with free sites, fixed order of cells, and fixed positions of cells in all other rows New iterative algorithm to improve the cell ordering within a given row Iterative row-based placement algorithm that applies single-row cell placement to each row in turn, with optional cell ordering improvement in the given row Average of 6.5% improvement in total wirelength

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Extensions Incorporate cell flipping into DP solution Linear programming formulation for Cell Ordering Problem Extend exact DP solution to k rows simultaneously Incorporate routability into objective function

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